Roots of Polynomials and The Derangement Problem

TL;DR

This paper introduces a novel proof of a classic probability result regarding permutations with no fixed points, using polynomial methods over finite fields and an independence argument.

Contribution

It provides a new, elementary proof connecting permutation fixed points to polynomial independence over finite fields, simplifying the understanding of the derangement probability.

Findings

Probability of no fixed points tends to e^{-1} as n grows large

New proof based on polynomial independence over finite fields

Simplifies understanding of classical derangement probability

Abstract

We present a new killing-a-fly-with-a-sledgehammer proof of one of the oldest results in probability which says that the probability that a random permutation on $n$ elements has no fixed points tends to $e^{-1}$ as $n$ tends to infinity. Our proof stems from the connection between permutations and polynomials over finite fields and is based on an independence argument, which is trivial in the polynomial world.